Coarse grid corrections in Krylov subspace evaluations of the matrix exponential

TL;DR

This paper introduces a coarse grid correction method to improve the efficiency of computing matrix exponential functions, leveraging multigrid techniques and splitting strategies for faster iterative evaluations.

Contribution

It proposes a novel coarse grid correction approach for matrix exponential evaluations, combining multigrid methods with vector splitting to enhance computational efficiency.

Findings

Significant speed-up in matrix exponential evaluations.

Effective in combination with Krylov subspace and Chebyshev methods.

Error estimates provided for multigrid variants.

Abstract

A coarse grid correction (CGC) approach is proposed to enhance the efficiency of the matrix exponential and $\varphi$ matrix function evaluations. The approach is intended for iterative methods computing the matrix-vector products with these functions. It is based on splitting the vector by which the matrix function is multiplied into a smooth part and a remaining part. The smooth part is then handled on a coarser grid, whereas the computations on the original grid are carried out with a relaxed stopping criterion tolerance. Estimates on the error are derived for the two-grid and multigrid variants of the proposed CGC algorithm. Numerical experiments demonstrate the efficiency of the algorithm, when employed in combination with Krylov subspace and Chebyshev polynomial expansion methods.